By "Algebraic Grothendieck group", do you mean the Grothendieck group of vector bundles?
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David SpeyerJun 11 '13 at 17:40

Yes , it is the Grothendieck group of vector bundles. The one of coherent sheaves is well-known.
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Al-AmraniJun 11 '13 at 17:51

It is mentioned in the above paper that it was the initial idea to construct such examples with huge $K_0$ among weighted projective spaces, but so far without success.
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Dietrich BurdeJun 11 '13 at 18:53

Yes, that is right. But recently, Adam Massey ( KH-Theory of Complete Simplicial Toric Varieties and Algebraic K-theory of Weighted Projective Spaces ) obtained some progress in very particular case of weights, that is (1,...,1,q).
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Al-AmraniJun 11 '13 at 19:25

2

In other words, the question is an open problem that looks well known among the people that know this sort of problems. Notice there is a minisection in the FAQ mathoverflow.net/faq about this.
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Mariano Suárez-Alvarez♦Jun 11 '13 at 19:40